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Publications

• "Chen-Khovanov spectra for tangles" with Tyler Lawson and Sucharit Sarkar. Michigan Mathematical Journal, accepted.
  • on the arXiv: arXiv:1909.12994.
• "Khovanov homotopy type, Burnside category, and products" with Tyler Lawson and Sucharit Sarkar. Geometry & Topology, accepted.
  • on the arXiv: arXiv:1505.00213.
• "Correction to the paper 'A flexible construction of equivariant Floer homology and applications'" with Kristen Hendricks and Sucharit Sarkar. Journal of Topology, accepted.
  • on the arXiv: arXiv:1810.03049
• "Spatial refinements and Khovanov homology" with Sucharit Sarkar. Proceedings of the International Congress of Mathematicians (ICM 2018), pp. 1153-1173 (2019).
  • on the arXiv: arXiv:1709.03602.
  • This expository article was not peer reviewed.
• "Bordered Floer homology and incompressible surfaces" with Akram Alishahi. Annales de l'Institut Fourier, accepted.
  • on the arXiv: arXiv:1708.05121.
• "Involutive bordered Floer homology" with Kristen Hendricks. Transactions of the American Mathematical Society, Vol. 372 (2019) No. 1, 389-424.
  • on the arXiv: arXiv:1706.06557.
• "Relative Q-gradings from bordered Floer theory" with Peter Ozsváth and Dylan Thurston. Michigan Mathematical Journal, Vol. 67 (2018) No. 4, 827-838.
  • on the arXiv: arXiv:1211.6990.
• Cornered Heegaard Floer homology with Christopher Douglas and Ciprian Manolescu. Memoirs of the American Mathematical Society, Vol. 262, No. 1266 (2019).
  • on the arXiv: arXiv:1309.0155.
• "A flexible construction of equivariant Floer homology and applications" with Kristen Hendricks and Sucharit Sarkar. Journal of Topology, Vol. 9 (2016) No. 4, 1153-1236.
  • on the arXiv: arXiv:1510.02449.
  • Correction: arXiv:1810.03049.
  • Review on MathSciNet (requires subscription).
• Bordered Heegaard Floer homology with Peter Ozsváth and Dylan Thurston. Memoirs of the American Mathematical Society, Vol. 254, No. 1216 (2018).
  • On the arXiv: math/0810.0687.
  • Review on MathSciNet (requires subscription).
• "The cube and the Burnside category" with Tyler Lawson and Sucharit Sarkar. To appear in Categorification in Algebra, Geometry and Physics, Contemporary Mathematics, American Mathematics Society.
  • on the arXiv: arXiv:1505.00512.
• "Heegaard Floer Homologies: lecture notes" in Lectures on quantum topology in dimension three, Panoramas et synthèses 48 (2016), Société Mathématique de France, 131-174.
  • On the arXiv: arXiv:1411.4540.
• "Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree" with Peter Ozsváth and Dylan Thurston. Journal of Topology, Vol. 9 (2016) no. 2, 607-686.
  • on the arXiv: arXiv:1404.2894.
  • Review on MathSciNet (requires subscription).
• "Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers" with David Treumann. Journal of the European Mathematical Society, Vol. 18 (2016), no. 2, 281-325.
  • On the arXiv: arXiv:1203.2963.
  • Review on MathSciNet (requires subscription).
• "Khovanov homotopy types and the Dold-Thom functor" with Brent Everitt, Sucharit Sarkar and Paul Turner. Homology, Homotopy and Applications, Vol. 18 (2016), no. 2, 177-181.
  • On the arXiv: arXiv:1202.1856.
• "Bordered Floer homology and the spectral sequence of a branched double cover I" with Peter Ozsváth and Dylan Thurston. Journal of Topology, Vol. 7 (2014), no. 4, 1155-1199.
  • On the arXiv: arXiv:1011.0499.
  • Review on MathSciNet (requires subscription).
• "Computing HF^ by factoring mapping classes" with Peter Ozsváth and Dylan Thurston. Geometry & Topology, Vol. 18 (2014), no. 5, 2547-2681.
  • On the arXiv: arXiv:1010.2550.
  • Review on MathSciNet (requires subscription).
• "Bimodules in bordered Heegaard Floer homology" with Peter Ozsváth and Dylan Thurston. Geometry & Topology, Vol. 19 (2015), no. 2, 525-724.
  • On the arXiv: arXiv:1003.0598.
  • Review on MathSciNet (requires subscription).
• "Notes on bordered Floer homology" with Peter Ozsváth and Dylan Thurston. In Contact and Symplectic Topology, Bolyai Society Mathematical Studies, Vol. 26 (2014) 275-355.
  • On the arXiv: arXiv:1211.6791.
  • Review on MathSciNet (requires subscription).
• "On transverse invariants from Khovanov homology" with Lenhard Ng and Sucharit Sarkar. Quantum Topology, Vol. 6 (2015), no. 3, 475-513.
  • On the arXiv: arXiv:1303.6371.
  • Review on MathSciNet (requires subscription).
• "A Steenrod square on Khovanov homology" with Sucharit Sarkar. Journal of Topology, 7 (2014), no. 3, 817-848.
  • On the arXiv: arXiv:1204.5776.
  • Review on MathSciNet (requires subscription).
• "A refinement of Rasmussen's s-invariant" with Sucharit Sarkar. Duke Mathematical Journal, Vol. 163 (2014), no. 5, 923-952.
  • On the arXiv: arXiv:1206.3632.
  • Review on MathSciNet (requires subscription).
• "A Khovanov stable homotopy type" with Sucharit Sarkar. Journal of the American Mathematical Society, Vol. 27 (2014), 983-1042.
  • On the arXiv: arXiv:1112.3932.
  • Review on MathSciNet (requires subscription).
  • Former titles: "A Khovanov homotopy type or two"; using an idea of Oleg Viro we can now show the two variants agree. "A Khovanov homotopy type"; the referee suggested adding the word "stable" to avoid confusion.
• "Errata to 'A cylindrical reformulation of Heegaard Floer homology'". Geometry & Topology, Vol. 18 (2014) 17-30.
  • on the arXiv: arXiv:1301.4919.
  • Review on MathSciNet (requires subscription).
  • Further minor corrections and explanations. Last updated December 16, 2015.
• "A faithful linear-categorical action of the mapping class group of a surface with boundary" with Peter Ozsváth and Dylan Thurston. Journal of the European Mathematical Society, Vol. 15 (2013), no. 4, pp. 1279-1307.
  • On the arXiv: arXiv:1012.1032.
  • Review on MathSciNet (requires subscription).
• "Heegaard Floer homology as morphism spaces" with Peter Ozsváth and Dylan Thurston. Quantum Topology, Vol. 2 (2011), no. 4, pp. 381-449.
  • On the arXiv: arXiv:1005.1248.
  • Review on MathSciNet (requires subscription).
• "A tour of bordered Floer theory" with Peter Ozsváth and Dylan Thurston. Proceedings of the National Academy of Science, Vol. 108, no. 20, pp. 8085-8092. May 17, 2011.
  • On the arXiv: math/1107.5621.
  • Review on MathSciNet (requires subscription).
• "Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology" with Peter Ozsváth and Dylan Thurston. Proceedings of 15th Gökova Geometry-Topology Conference, 91-119.
  • On the arXiv: math/0810.0695.
  • Review on MathSciNet (requires subscription).
• "Heegaard Floer homology, double points and nice diagrams." Proceedings of the Conference "New Perspectives and Challenges in Symplectic Field Theory, Stanford, June 2007." CRM Proceedings & Lecture Notes, Vol. 49 (2009), 327–342.
  • This paper in PDF format.
  • Review on MathSciNet (requires subscription).
• "Combinatorial cobordism maps in hat Heegaard Floer theory" with Ciprian Manolescu and Jiajun Wang. Duke Math. J., Vol. 145 (2008), no 2., pp. 207-247.
  • On the arXiv: math/0611927.
  • Review on MathSciNet (requires subscription).
• "Covering spaces and Q-gradings on Heegaard Floer homology" with Dan Lee. J. Symplectic Geom., Vol. 6 (2008), no. 1, 33-59.
  • On the arXiv: math/0608001. An updated version is available here.
  • Review on MathSciNet (requires subscription)
• "A Heegaard-Floer invariant of bordered 3-manifolds." Ph.D. thesis, Stanford University, 2006.
  • This paper in PDF format.
  • This paper in Postscript format.
  • Also available from ProQuest dissertation database (requires subscription)
• "A cylindrical reformulation of Heegaard Floer homology". Geom. Topol. 10 (2006), pp. 955-1097.
  • on the arXiv: math/0502494.
  • Review on MathSciNet (requires subscription).
  • Errata (listed above).
  • Further minor corrections and explanations. Last updated June 4, 2020.

Preprints (not yet accepted for publication)

• "A simplicial construction of G-equivariant Floer homology" with Kristen Hendricks and Sucharit Sarkar.
  • on the arXiv: arXiv:1609.09132.
  • Last updated: May 30, 2018.
• "Khovanov spectra for tangles" with Tyler Lawson and Sucharit Sarkar.
  • on the arXiv: arXiv:1706.02346.
  • Last updated: September 18, 2019.
• "Khovanov homology also detects split links" with Sucharit Sarkar.
  • on the arXiv: arXiv:1910.04246.
  • Last updated: October 9, 2019.

Papers written by undergraduate students I have advised are here.

Programming

• An introduction to Bohua Zhan's bfh_python.
• A technology demonstration (i.e., far pre-beta version) of a package to compute Heegaard Floer invariants using bordered Floer homology is here.
  • You will need Sage to run it.
  • This is version 0.2 (August 5, 2011), the second release on the web. It corrects several (critical) bugs found by Bohua Zhan.
  • This version contains new installation options, thanks to Nathan Dunfield. See the README (also courtesy of Dunfield).
  • Bohua Zhan has ported this program to C++. The port is much faster. Send him e-mail if you are interested in it.
  • Bohua Zhan also has a Sage rewrite and extension of the code, available on GitHub.
  • Partial documentation is included, in a PDF called BordProgDocs
  • Please e-mail me if you find bugs. (I'm sure they're there.) Programming style criticism also accepted.
  • Copyright 2010-2012 Robert Lipshitz, Peter Ozsvath and Dylan Thurston.
  • This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
• An extension of part of the bordered Floer package was used to perform computations for "Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers".
  • The extension is available here.
  • This is version 1 (March 14, 2012).
  • Like the paper, this extension is joint work with David Treumann.
  • Partial documentation is included.
  • The extension is Copyright 2012 Robert Lipshitz and David Treumann.
  • This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
• Code for computing the actions of Sq¹ and Sq² on Khovanov homology, written mainly by Sucharit Sarkar. Used in and based on our paper "A Steenrod square on Khovanov homology".
• Code for computing the refined s-invariant from Sq² on Khovanov homology, written mainly by Sucharit Sarkar. Used in and based on our paper "A refinement of Rasmussen's s-invariant".

Some programs by Columbia-affiliated topologists are here. CompuTop has a more extensive list of programs for doing computations in low-dimensional topology.

Support

My research has been supported by an NSF Graduate Research Fellowship, an NSF Mathematical Sciences Postdoctoral Research Fellowship, a Sloan Research Fellowship, and NSF Grants DMS-0905796, DMS-1149800, DMS-1560783, and DMS-1642067. Specific research citations are listed in the papers themselves.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.